Identification of a reversible quantum gate: assessing the resources

Abstract

We assess the resources needed to identify a reversible quantum gate among a finite set of alternatives, including in our analysis both deterministic and probabilistic strategies. Among the probabilistic strategies, we consider unambiguous gate discrimination-where errors are not tolerated but inconclusive outcomes are allowed-and we prove that parallel strategies are sufficient to unambiguously identify the unknown gate with minimum number of queries. This result is used to provide upper and lower bounds on the query complexity and on the minimum ancilla dimension. In addition, we introduce the notion of generalized t-designs, which includes unitary t-designs and group representations as special cases. For gates forming a generalized t-design we give an explicit expression for the maximum probability of correct gate identification and we prove that there is no gap between the performances of deterministic strategies and those of probabilistic strategies. Hence, evaluating of the query complexity of perfect deterministic discrimination is reduced to the easier problem of evaluating the query complexity of unambiguous 4 Identifying an unknown unitary evolution, available as a black box, is a fundamental problem in quantum theory [1][2][3][4][5][6][7][8][9], with a wide range of applications in quantum information and computation. In quantum computation, the problem is known as oracle identification [10][11][12][13][14] and is the core of paradigmatic quantum algorithms such as Grover's [15] and . In quantum information processing, the identification of an unknown unitary gate plays a key role in the stabilizer formalism of quantum error correction [17,18] and in its generalization to unitary error bases [19][20][21][22][23], in the security analysis of quantum cryptographic protocols that encode secret data into unitary gates [24][25][26][27][28][29], New Journal of Physics 15 (2013) 103019 (http://www.njp.org/)

Unitary channel discrimination beyond group structures: Advantages of sequential and indefinite-causal-order strategies

“…The first contribution of our work is precisely to show examples of discrimination tasks of unitary channels in which sequential strategies are advantageous, when compared to parallel strategies that allow the same number of copies. In fact, contrarily to the tasks of error-free and unambiguous unitary channel discrimination [21], we show that sequential strategies can achieve perfect discrimination, while parallel cannot.…”

“…[20] showed once more that, for any number of copies, sequential strategies are not advantageous when compared to parallel strategies. For related tasks such as error-free and unambiguous unitary channel discrimination [21], unitary estimation [20], unitary learning [22], and unitary store-and-retrieve [23], parallel strategies were also proven to be optimal. Up to this point, no unitary channel minimum-error discrimination task in which sequential strategies outperform parallel strategies are known, to the extent of our knowledge.…”

Multiple-shot and unambiguous discrimination of von Neumann measurements

“…One of the first results was the study of discrimination of unitary operators [15,16]. Later, this has been extended to various settings, such as multipartite unitary operations [17] and the case of discrimination among more than two unitary channels [18]. In the work [19] the authors formulated necessary and sufficient conditions under which quantum channels can be perfectly discriminated.…”

“…is a three-outcome measurement on C d ⊗C d that satisfies constrains(18) end σ is a state on the extended Hilbert space C…”

Optimal entanglement-assisted discrimination of quantum measurements

“…More recently, this concept has been extended to discrimination of quantum operations [17][18][19][20][21][22][23][24][25][26][27][28] and measurements [29][30][31]. While sharing many similarities with discrimination of quantum states, discrimination of quantum devices admits intriguing novel strategies and phenomena [32][33][34][35][36][37][38] such as using probes entangled with auxiliary systems, or the perfect distinguishability of any two unitary operations when a sufficiently large but finite number of copies of the operation is available [17].…”